I'm looking at an example from a book I'm reading,
How does it formulate the mean?
I am thinking $$E[X[n]] = \sum\limits^\infty_{-\infty} X[n]P[X[n]] =\\
\sum\limits^\infty_{n=-\infty} X[n = even] \...

I'm trying to set up a predator-prey model, where the predator affects the distribution of the prey. Effectively, over time the prey has moved from areas of high concentration of the predator to areas ...

I have the following problem: Let's assume $G$ is a graph with vertices in red or blue colour. There is no limitation on how we connect the vertices, i.e., a red vertex can be connected with either ...

$A$ starts with $i$ coins, $B$ with $N-i$. At each trial, $A$ gives one coin to $B$ with probability $p$ or $B$ gives one coin to $A$ with probability $q$ where $p+q=1$.
This can be modeled as a 2D ...

I'm reading Ross' "Stochastic Processes" book, in the part of random walks, and I have troubles to understand an argument.
Consider $\{S_n\}_{n\ge 0}$ the simple random walk, with $\mathbb{P}\left(X=...

Consider $(X_n)_{n\ge 1}$ iid with $P(X_1=1)=P(X_1=-1)=\frac{1}{2}$. Then there is the sumprocess $S_n:=\sum_{i=1}^n X_i$ and the stopping times $$T_{a,b}:=\min\{n\ge 1:S_n\in\{a,b\}\},$$ $$T_{a}:=\...

I am learning about algorithms for finding subgraphs with high mean escape time, and I am wondering if someone could enlighten me on what applications there are for such a task. Applications to either ...

This is kind of a vague question, but I will try to make it as precise as possible. So I have been studying random walks and their properties, in particular, I am interested in conditions that imply ...

The "random walk" process is well known for mathematicians, see for instance https://en.wikipedia.org/wiki/Random_walk . It is also known as "the drunk walk".
It is demonstrated that if you have a n ...

There's a gambler with initial money $k=100\$$. He plays till bankruptcy or till having $N=500\$$. In every game he wins $100\$$ with probability $p=\frac{1}{2}$, loses $100\$$ with probability $q=\...

Let (Sn)n∈N be a random walk process with increments that are independent.
The value of the random walk increase by one in one time step with probability
p and decrease by one in one time step with ...

Suppose $(x_t)_{t=0}^\infty$ is a random starting from $x_0=0$ with transition probability $P(x_{t+1}-x_t)=p\mathbf 1(x_{t+1}-x_t=1)+q\mathbf 1(x_{t+1}-x_t=-1)$ where $p\ge0,q\ge0,p+q=1$. We are given ...

Assume a person random walker takes equal steps to the right or left with equal probability.
Probability that taking n steps, the person walking will be displaced 1 standard deviation or greater in ...

A set $G$ endowed with an associative binary operation is called a semigroup if it possesses an identity element.
Thus a semigroup is short of a group in that it may not be closed under inverses.
Let ...

I am reading this paper related to an algorithm for nonsmooth optimization problems. After many simplifications, I was able to formalize the method as follows: let $\Bbb B $ denote the unit ball in $\...

Consider a random walk on a line of integers. Suppose we start from the state $x$. Then, the probability of jumping from $x$ to $x+1$ is $p(x, x+1)=p$, and the probability of jumping from $x$ to $x-1$ ...